Slice the onion into thin rings and heat olive oil in a skillet over medium heat. Cook onion rings until golden brown and caramelized, about 20-25 minutes.

In a small bowl, mix mayonnaise and horseradish, then season with salt and pepper. Cover and refrigerate.

Fry bacon in a separate pan until crispy, about 10-15 minutes. Transfer to a paper towel-lined plate.

In a mixing bowl, combine ground beef, minced garlic, breadcrumbs, beaten egg, salt, and pepper. Mix gently and form into four patties.

Preheat grill to high heat. Cook patties for about 6 minutes per side, aiming for an internal temperature of 165°F.

Melt butter in a skillet and toast buns cut-side down until golden brown, about 2-3 minutes.

Assemble burgers by spreading horseradish mayo on the bottom bun, adding a patty, bacon, and fried onions, then topping with the other bun.